Supplementary Information: Enhancement of the

“tractor-beam” pulling force on an optically

bound structure

Jana Damkova, Lukas Chvatal, Jan Jezek, Jindrich Oulehla, Oto

Brzobohaty and Pavel Zemanek∗

Institute of Scientific Instruments of the CAS, v.v.i.,

Kralovopolska 147, 612 64 Brno, Czech Republic

1 Comparison of the measured trajectories with theoretical

calculations

In Figures S1 and S2 we compare more experimental and theoretical results related to the behavior

of a particle pair optically bound in the tractor-beam. The measured trajectories marked with

zigzag curves are compared with the calculated velocities of the stable optically bound particle pairs.

We observed very good agreements between inter-particle distances and calculated stable positions.

These results follow the conclusions of the main text regarding the particle motion in the second/third

scattering lobe (Figures S1a-c). Here the interaction force due to the presence of the first particle

rules the particle movement and thus, there are significant changes in the direction of particle pair

motion. Figure S1d demonstrates such reversal of particle pair motion in different lobe, too. In a

large number of measurements, the particle pair was optically bound in higher-order scattering lobes,

where the interaction force was not dominant and thus the pair was pulled by the S-polarized tractor-

beam (see Figure S2) as an isolated particle. Even here the inter-particle distances correspond to

the calculated stable positions and the second particle was moving generally close to the scattering

lobe maxima.

1

5 100

5

10

15

20

25

30

35

∆z /λm

∆y/λ

m

0 5 10∆z /λm

0 5 10∆z /λm

0 5 10∆z /λm

40

0

5

10

15

20

25

30

35

∆y/λ

m

40

45

50

55

15

b da b ca c

0

Pushing

Pulling

Figure S1: Comparison of the measured trajectories with the calculated particle velocities. The gray

thin curves represent scattering-lobe maxima. Calculated velocities of the particle pair along the z–

axis are encoded in the length of the triangles (proportional to√vpair). For comparison the velocity

of an isolated particle visol is shown at the center of particle 1 placed at the origin of the system

of coordinates. The solid zigzag curves represent experimental results and their colors encode the

direction of the particle pair motion. (λm = 400 nm, incident angle of the S-polarized tractor-beam

α = 2.15 ◦, polystyrene particles with 820-nm diameter).

2

0 5 10∆z /λm

0

5

10

15

20

25

30

35

∆y/λ

m

40

45

50

55

15 0 5 10∆z /λm

15 0 5 10∆z /λm

15 0 5 10∆z /λm

15

dba cPushing

Pulling

Figure S2: Comparison of the measured trajectories with the calculated particle velocities. The gray

thin curves represent scattering-lobe maxima. Calculated velocities of the particle pair along the z–

axis are encoded in the length of the triangles (proportional to√vpair). For comparison the velocity

of an isolated particle visol is shown at the center of particle 1 placed at the origin of the system

of coordinates. The solid zigzag curves represent experimental results and their colors encode the

direction of the particle pair motion. (λm = 400 nm, incident angle of the S-polarized tractor-beam

α = 2.15 ◦, polystyrene particles with 820-nm diameter).

3

2 Theoretical description of electric-field extremes in scat-

tering lobes

Let us assume a simplified geometry of two interfering plane waves whose wavevectors form an angle

π − 2α. This geometry ignores the reflection at the mirror but it is equivalent to the geometry used

in the experiment for particles placed far from the mirror. Let us denote

kx = 0, (1)

ky = k cosα, (2)

kz = k sinα, (3)

where k = 2π/λm, α and λm have the same meaning as in the main text. Assuming the wavevectors

pointing against the z -axis (see wavevectors k1 and k2 in Figure 1a in the main text)

k1 = [0,−ky,−kz] = [0,−k cosα,−k sinα], (4)

k2 = [0,+ky,−kz] = [0,+k cosα,−k sinα], (5)

we can write for the interference incident field of the S-polarized tractor-beam:

Exi = E0 exp[i(−kzz + kyy)] + E0 exp[i(−kzz − kyy)] = 2E0 cos(kyy) exp(−ikzz). (6)

For a general direction θ in the zy-plane (tan θ = z/y) and few wavelengths away from the scattering

particle, the scattered field is well described as a propagating spherical wave

Exs = E0FS(θ)

krexp[i(kr + φS(θ)], (7)

where real functions FS and φS characterize the scattering of both incident plane waves. The total

electric field energy density yields

|E|2 = |Exi + Exs|2

= 4|E0|2 cos2(kyy) + 4|E0|2 cos(kyy) cos[kzz + kr + φS(θ)]FS(θ)

kr+ |E0|2

(FS(θ)

kr

)2

≈ 4|E0|2 cos(kyy)

[cos(kyy) + cos[kr + φS(θ) + kzz]

FS(θ)

kr

].

(8)

Let us set the particle position to the maximum of the fringe along the y-axis assuming kyy = πn with

an integer n. The scattering amplitude factor FS(θ) is positive and changes slower with a position

than the cosine terms. We may therefore state the condition for the constructive interference of the

two terms as

kr + φS(θ) + kz sinα = πm, (9)

where m is another integer, with even/odd values corresponding to the constructive/destructive

interference. Let us further assume ‘effectively isotropic’ scattering represented by the parameter

b = φeff/π. (10)

4

Using (kr)2 = k2(z2 + y2) = (kz)2 + (πn/ cosα)2, we solve for the coordinate z and obtain

z =λm

2 cos2 α

[(b−m) sinα±

√(m− b)2 − n2

]. (11)

Going to either side from the main forward-scattering axis, we index the consecutive lobes of de-

creasing intensity by integer index p. At the in-fringe maxima p = 0, 2, 4, 6 . . . and at minima

p = 1, 3, 5, . . .

A comparison with the exactly calculated field distribution revealed that the value φeff = π/2 gives

a very good coincidence if the maxima/minima corresponding to a given lobe lie on curves defined

by Eq. (11) and a constriction m = n+ p is valid. It gives

zp =λm

2 cos2 α

[(b− n− p) sinα±

√(2n+ p− b)(p− b)

], (12)

which was used to draw the electric field density maxima in Figures 2a, 2b, 3a, 4b in the main text.

The derivation of the P-polarized tractor-beam is analogical; for the external field of the tractor-beam

we can write

Ee = E0

0

+ sinα

cosα

exp[i(−kzz + kyy)] + E0

0

− sinα

cosα

exp[i(−kzz − kyy)]

= 2E0 exp(−iz sinα)

0

i sin(kyy) sinα

cos(kyy) cosα

(13)

and the corresponding scattered field reads

Es = E0

0

− sin θ

cos θ

FP(θ)

krexp[i(kr + φP(θ))]. (14)

The field energy density, after neglecting the term ∝ F 2P, is

|E|2 ≈ 4E20

[sin2(kyy) sin2 α + cos2(kyy) cos2 α

]+ 4E2

0

FP(θ)

kr[cos(kyy) cosα cos θ cos Φ− sin(kyy) sinα sin θ sin Φ] ,

where

Φ = kr + φP(θ, α) + kz sinα. (15)

The small value of α reduces the terms with sinα. Also, for θ in the narrow interval of interest,

φP(θ) ≈ φS(θ) and we may well use the arguments for the case of S-polarization and employ Eq.

(12).

5

3 Stable configurations of optically bound pairs

During calculation of optical forces, we solve the electromagnetic scattering problem in a self-

consistent way, i.e., we take into account enough reflections to achieve numerical convergence. If

the spheres are separated by several wavelengths, multiple scattering between spheres of a degree

higher than 2 contributes negligibly to the total optical force. Thus the optical force acting on

sphere 2 is determined mainly by the interaction with the field scattered only once from sphere 1

and superposed with the external incident tractor-beam field.

Let us assume that sphere 1 is held fixed in a fringe, generally with some external nonzero force.

It turns out that the stable positions of sphere 2 (i.e. static equilibrium, where F2,z = 0) emerge

preferentially very close to the maxima of the scattering lobes formed by the incident field and the

field scattered by sphere 1 alone, as illustrated by the green markers in Figures S3 almost overlapping

with white curves denoting the intensity maxima of the lobes (following Eq. 12). Conversely, if sphere

2 is held fixed in a fringe, sphere 1 would take up some equilibrium position close to the intensity

maxima arising from the light scattered by sphere 2. An intuitive picture of this behavior is illustrated

in Figures 2c-h in the main text, where we show the dependence of the optical force acting on one

particle on the external electric field profile and the particle position while the second particle is

held fixed. However, with respect to the mirror symmetry in the xy-plane, the conditions for the

static equilibrium (i.e. F1,z = F2,z = 0 giving v1,z = v2,z = 0) can be fulfilled simultaneously for both

spheres only for the counter-propagating plane waves, i.e., α = 0, kz = 0.

Any nonzero α > 0 breaks the aforementioned symmetry and stable configurations with constant

inter-particle distances arise only if both spheres move uniformly with the same velocity, implying

equal forces acting upon both particles F2,z = F1,z. This case is presented in Figures S3 in the

form of orange (pushing) and blue (pulling) dots for the S-polarized tractor-beam. Comparing green

and orange/blue dots one finds that there are much more stable moving configurations of the pair of

particles. These configurations are mainly in the intensity maxima of the lobes but there are also some

exceptions violating this rule. This behavior also indicates that the moving stable configurations arise

as a balance between mutual demands of each particle to reach the intensity lobe maxima created

by the other sphere. Figures S4 compare the pushing and pulling behavior of the moving stable

configurations for both S- and P-polarizations and different incident angles α.

4 Hydrodynamical interaction between particles in a mov-

ing pair

The observed motion of a pair of particles is slow enough to give low Reynolds number Re ≈10−6 − 10−4. Thus the inertial terms can be omitted in appropriate equations of motion and the

6

S-polarization, α = 0 ◦ S-polarization, α = 0 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

S-polarization, α = 3 ◦ S-polarization, α = 3 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

S-polarization, α = 1 ◦ S-polarization, α = 1 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

S-polarization, α = 4 ◦ S-polarization, α = 4 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

S-polarization, α = 2 ◦ S-polarization, α = 2 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

S-polarization, α = 5 ◦ S-polarization, α = 5 ◦

F 2 = 0 F 2 − F 1 = 0

-5

0

5

10

15

20

25

30

35

40

∆y/

(λm/

cosθ)

∆z /λm

-15 -10 -5 0 5 10 15

∆z /λm

-15 -10 -5 0 5 10 15

Figure S3: Comparison of static (green) and moving stable (orange and blue) configurations of a

pair of particles for different incident angles α. Green dots denote the stable position of particle 2

(i.e. F2,z = 0) if particle 1 (light larger blue dot) is kept fixed at maximum of the interference fringe

along the y-axis. Orange and blue dots denote stable positions of particle 2, where both particles

move with the same velocity, i.e., F2,z = F1,z and F2,y = F1,y. The background map shows electric

field energy density |E|2 (in log scale) as in Figure 3a of the main text. White and gray curves mark

the intensity maxima and minima of |E|2 following the extrema of the lobes of the scattered field

done by Eq. (12). The white annulus around particle 1 indicates the range, where the particles are

too close to each other to plot the data properly.

7

S-polarization, , =0.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =0.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

S-polarization, , =3.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =3.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

S-polarization, , =1.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =1.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

S-polarization, , =4.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =4.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

S-polarization, , =2.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =2.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

S-polarization, , =5.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

"y=6

SW

0

5

10

15

20

25

30

35

40

P-polarization, , =5.00/

F2;z ! F1;z = 0

"z=6m

0 5 10 15

Figure S4: Theoretical results of pushing (orange) and pulling (blue) behavior of the moving stable

pairs of particles for both S- and P-polarizations and different incident angles α (other parameters

same as mentioned in the main text).

8

particle velocities are proportional to the forces as(v1

v2

)=

(µ11 µ12

µ21 µ22

)·

(F1

F2

), (16)

where v1,2 denotes the velocity of each particle, F1,2 corresponds to external forces acting on the

particles (in our case exclusively given by the optical forces), and µab are 3 × 3 mobility tensors.

These tensors can be split into a longitudinal and transverse component1 as

µab,ij = (µ⊥)ab,ij(E− P) + (µ‖)ab,ij P , (17)

where E is the identity matrix, P = nTn is the projection matrix, n is the unit vector in a direction

of a sphere connecting line, i, j ∈ {x, y, z} denote the coordinates and a, b ∈ {1, 2} index of the

particle. The sphere centers are separated by a center-to-center distance r = rn.

Assuming spheres of the equal radius a, the normalized tensor components expressed up to the

seventh order in t = a/r 1 are the following

µ‖11/µ0 = 1− 15

4t4 +

13

4t6,

µ⊥11/µ0 = 1− 17

16t6,

µ‖12/µ0 =

3

2t− t3 +

75

4t7,

µ⊥12/µ0 =

3

4t+

1

2t3,

(18)

where µ0 = 1/6πηa is the Stokes mobility of an isolated sphere and η the medium viscosity. The re-

maining components follow the relations µ11 = µ22, and µ12 = µ21. The condition for the sustainable

uniform motion of the pair v1 = v2 ≡ v implies F1 = F2 ≡ F and

v = (µ11 + µ12)F ≡MF, (19)

where we introduced an auxiliary ‘pair-mobility’ tensor M.

Let us assume the particle pairs are constrained to the yz-plane and their orientations are parametrized

by an angle β, where n = (0, cos β, sin β). The velocity with a just single nonzero component along

z requires nonzero Fz as well as Fy, given by equations

vz = (M‖ sin2 β +M⊥ cos2 β)Fz + sin β cos β(M‖ −M⊥)Fy,

vy ≡ 0 = sin β cos β(M‖ −M⊥)Fz + (M‖ cos2 β +M⊥ sin2 β)Fy.(20)

Substituting for Fy we obtain

vz =

[M‖ sin2 β +M⊥ cos2 β +

[(M‖ −M⊥) sin β cos β]2

M‖ cos2 β +M⊥ sin2 β

]Fz. (21)

Figure S5 compares velocities of pairs with different inter-particle distances r and orientations β with

respect to the y-axis. Keeping the mobility components only up to the first degree in t (using Oseen

approximation) yields

vz ≈[1 +

3

4(1 + cos2 β)t+

9

16

[sin β cos β]2

1 + 34(1 + cos2 β)

t2]vz0, (22)

9

0 10 20 30 40 50 60 70 80 90

1

1.1

1.2

1.3

1.4

1.5

1.6r / a = 2

r / a = 4

r / a = 8

r / a = 16

r / a = 32

v z/v z

0

β (◦)

Figure S5: Speed vz of a pair of particles of the radius a related to the speed vz0 of a pair with

ignored hydrodynamic biding as a function of the inter-particle distance r and pair orientation β

with respect to the y-axis. The dashed curves show the effect of disregarding the ‘constraint’ term

in Eq. (21). The Ossen approximation in Eq. (22) is shown by dotted curves and it overestimates

mobility for the particle in contact, but for r/a > 8 approximately overlaps with the higher-order

result. The value vz/vz0 = 1 corresponds to the case, where the hydrodynamic binding is ignored.

where we denoted vz0 = µ0Fz as the velocity of the pair omitting the hydrodynamic interaction.

Note that the second order term t2 is a consequence of the velocity constraint (Fy 6= 0). Even though

this term vanishes for orientations β = 0, β = π/2, it contributes by a small amount for any other

inclination. It implies that in the moving stable state, both spheres must be slightly deflected out of

the fringe centers in the same direction of the y-axis.

For the particle size used in the presented experiments, the fringe stiffness would allow only a tiny

fringe-transverse displacement so that the fringe-parallel forces remain almost unaffected. However,

for spheres positioned close to each other and sphere sizes weakly sensitive to the standing wave

fringes along the y-axis, the hydrodynamic interaction together with the binding forces may cause

significant shift of the spheres out of the equilibrium position along the y-axis.

References

1 Reuland P, Felderhof BU, and Jones RB. Hydrodynamic interaction of two spherically symmetric

polymers. Physica A 1978; 93: 465–475.

10